Devn 



IV - EQUATION OF ROLL MOTION - 



The roll equation is obtained from the pitch equation by- 

 removing the forcing term located in the right hand side of the 



equation 



(II -4.1) l Ixx+Jxx " 



. 2 a 



»myf 



1 d<p 



|d_f Npp — + mg 



m+riiyy/d h 2 



dt 



r+a+ g(s) 



H 



f =. ° 



with 



xx 



m 



yy 



= j 

 yy 



= m „ and m 



and Nff = N Q „ 



= m 

 yy xx 



Experiments have shown (cf. Section I) that the roll am 

 plitude may be much higher than the pitch amplitude . This fact 

 suggest the introduction in the equation (II-4. l) of a new viscous 

 damping term . This term is obtained using the Morison O'Brien 

 equation which gives the force d F (z) acting on a strip of length dz 

 for an infinite length cylinder in an unstationary flow [27J 



p , TTD 2 d U(t) 



TTD 

 — C D .D |u(t)|.U (t) + C M P — 



d t 



where D is the cylinder diameter , U (t) the velocity of the fluid 



relative to the cylinder , CI and C A , are constants 



DM 



The second term of this equation was already taken into 

 account : the first one corresponds to a non-linear damping . 

 Equation (II-4. l) becomes 



(II-4.2) 



.Ixx+Jxx- 



*_yf \d (p (1) df 



i2 + n; 



m+myy/dt 



ff dt 





d<f 

 dt 



df 

 dt 



+mg 



zG 



r+a+3l~g(s) 



f = o 



with 



(1) 

 Nif>H> = Nff 



(2) 



Nvpf 



P 



T c^. D 



2 D 



r z 4 G (H- zG) 4 " 



with D = 2 R 



1038 



